Of Set Theory and (False?) Claims of Logical Contradictions

OK, starting with Neil Rickert and not ending with him, people have made bald claims that my methodology of set comparison wrt infinite sets and their cardinality, causes logical contradictions.
Unfortunately all alleged logical contradictions have been my opponents' inability to properly use my methodology.
So it is strange that I get these claims of logical contradictions and yet no one can say what those are.
And not only that no oen can say what the practical application is for saying that the set of all nonnegative integers and all positive integers have the same cardinality. It doesn't appear to have any use at all. Therefor it seems to me that saying the first set has a greater cardinality than the second set is not wrong becasue it has no effect whasoever.
So can anyone tell me what the alleged logical contradictions are or are all my opponents just full of shit?
OK, starting with Neil Rickert and not ending with him, people have made bald claims that my methodology of set comparison wrt infinite sets and their cardinality, causes logical contradictions.
Unfortunately all alleged logical contradictions have been my opponents' inability to properly use my methodology.
So it is strange that I get these claims of logical contradictions and yet no one can say what those are.
And not only that no oen can say what the practical application is for saying that the set of all nonnegative integers and all positive integers have the same cardinality. It doesn't appear to have any use at all. Therefor it seems to me that saying the first set has a greater cardinality than the second set is not wrong becasue it has no effect whasoever.
So can anyone tell me what the alleged logical contradictions are or are all my opponents just full of shit?
69 Comments:
At 8:06 AM, Rich Hughes said…
Give examples, don't talk in generalities. The obvious person with a problem is you, who couldn't understand set theory so you made your own one up....
At 8:09 AM, Joe G said…
Exactly! The people making the claim of logical inconsistencies need to give examples. However they cannot which makes their claims = bullshit.
And I understand set theory. There isn't anything that I said that upsets set theory.
At 8:30 AM, Rich Hughes said…
1. There are plenty if examples at TSZ
2. You clearly didn't understand set theory, you got basic questions wrong
3. You claim 'it's useless anyway' yet devote time to (looking stupid at) it.
At 8:48 AM, Joe G said…
1 No, there aren't any examples at TSZ. You are lying again, as usual
2 Your false accusation means nothing BTW there isn't anything "basic" about infinite sets.
3 No one can tell me any practical application
At 8:49 AM, Joe G said…
Let's see if Richie can go to TSZ and bring back one valid example that demonstrates a logical inconsistency. keiths tried and failed miserably.
My bet is Richie is full of shit, still.
At 1:43 PM, Unknown said…
If you want to understand Set Theory then study it. Take a course. And stop accusing people of lying 'cause you're too lazy to do some research.
At 1:48 PM, Unknown said…
You'd lose that bet if you bothered to go find out instead of just waiting for everyone else to do the work for you.
Again, practical applications are not the point. Practical applications are concerned with finite sets for the most part. And the logic is different.
At 2:06 PM, Joe G said…
I understand set theory. I just disagree with a little and obvioulsy insignificant part of it
I notice that you didn't produce any alleged logical inconsistencies either.
And practical applications is the point because that is how you can demonstrate an inconsistency.
At 2:08 AM, Unknown said…
No, I didn't bother to do the work to find some mathematical inconsistencies with your viewpoint. If you want to know what they are you do the work. If you're going to say things that run counter to all the work that's already been done and built upon it's up to you to see if you're right or wrong. The mathematical world is not going to take you by the hand and walk you through. It will, however, leave you to cry in the wilderness and look like a fool.
If you do understand Set Theory then explain to me how the Axiom of Choice and Zorn's Lemma are equivalent.
At 7:05 AM, Joe G said…
Fuck you Jerad. There aren't any inconsistencies.
And fuck you as nothing is built upon what I disagree with.
At 9:16 AM, Unknown said…
"And fuck you as nothing is built upon what I disagree with."
So, you never really did take a Set Theory course?
What mathematics courses have you taken?
You're the one making claims that run counter to mathematics. It's up to you to show where everyone else is wrong.
At 9:24 AM, Joe G said…
Anyone making the claim saying my claims run counter to mathematics had better demonstrate such a thing.
Right now my claims run counter to some people's claims. And strange that those people cannot show they are right...
At 9:25 AM, Joe G said…
Please tell me Jerad, what exactly is built on the premise that two infinite but countable sets have the same cardinality? Please be specific.
At 10:24 AM, Joe G said…
And not just any two sets. Tell me what is built on the premise that you can take two sets and in a finite world one has twice the number of elements as the other, make them infinite and all of a sudden they have the same number of elements.
What, exactly, is built on that?
At 2:59 PM, Rich Hughes said…
1. f(n) = −n
2. I can uderstand why you don't find them basic, you're not very gifted at math.
3. Then why do you care / are you devoting serious tard to it?
At 3:34 PM, Joe G said…
1 Nope, that doesn't contradict anything I have claimed. Again. you don't just get to baldly say something, especially when I had already covered it moron.
2 No one finds them basic
3 Because now it has been brought to my attention and it is entertaining watching small minded evoTARDs try to grasp a new concept.
At 3:49 PM, Joe G said…
So, to be clear there are no examples of any inconsistencies with my methodology. Two number line segments heading in the opposite direction? Just absolute the negative if you need thenm to run side by side that is if you are too stupid to be able to follow each of the trains if they head in the opposite direction.
Is this really the best you chumps have to offer?
Really?
And you assholes say that I am not gifted at math?
BWAAAAAAHHAHAHAAAAHAHAHAHAHAAAAAAAAHAHAHAHA
fucking cupcakes...
At 4:42 PM, Rich Hughes said…
1. Do you always absolute them? Smells like an internet tard making it up as he gooes along...
2. Oh, you've asked everyone? Thought not. More fattytard nonsense
3. the entertainment always has been, and always will be, You, Chubs. Google yourself.
At 6:13 PM, Joe G said…
1 No, that is only for tards who cannot follow two things going in opposite directions. Just as I said.
2 So far even everyone from your side sez to toss intuition out of the window. They have no clue how to deal with infinite sets except to throw up their hands and sau they must be the same.
3 Again, it makes me happy to know that you think so, cupcake.
At 6:18 PM, Joe G said…
So, to be clear there are no examples of any inconsistencies with my methodology.
And Richie has proven to be the equivalent of a mathematical limp noodle.
"Ohs noes!? Two otherwise equivalent lines heading in the opposite direction!!11!!! That confuses my pointy little head so Joe must be wrong!!11!!11!!!1!!!"
Nice job limpy...
At 7:25 PM, socle said…
Joe,
It looks like you have removed some of the inconsistencies in your system in certain cases, but what would your method tell you about the cardinalities of these sets?
A = {3/2, 5/3, 7/4, 9/5, 11/6, ...}
and
B = {4/2, 7/4, 10/6, 13/8, 16/10, ...}
All the numbers in A are between 1.5 and 2, while all the numbers in B are between 2 and 1.5. The elements in both sets are listed in increasing order.
And keep in mind that "defer to Cantor" is not an option, since your method is incompatible with his.
At 7:50 PM, Joe G said…
There aren't any inconsistencies. That is the set of imaginary things.
And are you admitting that you are too stupid to use my very simple method?
Really?
Also the numbers in the first set are not between 1.5 and 2 as the first number is 1.5. The same goes for the second set> ie not between 2 and 1.5 as the first element is 2.
OK seeing that 0 is out for both sets it has no involvement at all, we would start Train A at 1.5 and Train B at 2, with both trains starting with a 1 count in their sets and both heading in the same direction.
And then start counting.
But then again my methodology applied to sets going to infinity, not sets of infinite sizes.
Those are two separate entities.
At 8:23 PM, Rich Hughes said…
"But then again my methodology applied to sets going to infinity, not sets of infinite sizes."
So a set "going to infinity" doesn't have "infinite size"?
At 8:29 PM, Joe G said…
Yes it does. But not all sets of infinite size go to infinity.
Hence the difference, dumbass.
At 8:42 PM, Rich Hughes said…
You've contradicted yourself, Chubstard:
1. you claim you can't use your method for sets "of infinite size"
2. You agree sets "going to infinity" have "infinite size"
3. But you also claim you can use your method for sets "going to infinity"
Are you making a new tardlogic to match your new math?
At 8:43 PM, socle said…
Joe,
The trouble with this is that Cantor's method works fine for all these sets, including those that Winston listed, which have no obvious "starting point" or even any natural ordering. Based on what you have said, it appears your method only works for sets with the same distance between consecutive elements. If that is correct, then the Einstein Train method is very limited indeed.
I had expected that you would say that set A has greater cardinality, as the train for set A will eventually have arbitrarily more markers than the train for set B, at least until train B passes 1.5.
At 9:36 PM, Joe G said…
No, Richie, I just explained myself.
you claim you can't use your method for sets "of infinite size"
Of infinite size that do not go to infinity.
At 9:41 PM, Joe G said…
The trouble with this is that Cantor's method works fine for all these sets
There isn't any working or not working. There doesn't seem to be any practical use for saying what Cantor sez wrt.
What Cantor does is like saying "let all elements of every set = e. Then with sets of infinite size it's e,e,e,e,e,e,e,e,e,... all the way down!"
And I said what to do with my methodology, ie how to tell which has a greater cardinality. However it may differ depending on when it is being observed ie when the set tally is observed.
Take the train ride.
At 9:57 PM, Joe G said…
socle,
Thank you for trying to move the goalposts after you had seen that my methodology worked wrt what was being debated.
At 10:15 PM, socle said…
Thank you for trying to move the goalposts after you had seen that my methodology worked wrt what was being debated.
Goalposts? Cantor's method works for all sets, finite or infinite, and I assumed yours was intended to as well.
On the other hand, there are various definitions of "density" of subsets of the natural numbers (as opposed to cardinality) that are somewhat related to your idea, so looking at finite "pieces" of sets as you were doing does have applications.
At 2:19 AM, Unknown said…
"Just absolute the negative if you need thenm to run side by side"
:)
"So far even everyone from your side sez to toss intuition out of the window."
How is lining up the elements of two sets to see if they are the same size tossing out intuition?
"But then again my methodology applied to sets going to infinity, not sets of infinite sizes.
Those are two separate entities."
Just making it up as he goes along.
"Yes it does. But not all sets of infinite size go to infinity."
That is true. {½, ⅓, ¼ . . . . } is infinitly large but 'goes' to zero. Most mathematicians would say converge though.
"What Cantor does is like saying "let all elements of every set = e. Then with sets of infinite size it's e,e,e,e,e,e,e,e,e,... all the way down!""
Perhaps you should look at what Cantor was doing again.
"And I said what to do with my methodology, ie how to tell which has a greater cardinality. However it may differ depending on when it is being observed ie when the set tally is observed. "
Set tally? You mean like counting how many things are in each set? Really? Gosh, that sounds like what I want to do!! Take an infinite set, see if you can line it up with the positive integers. If you can then the mystery set has the same cardinality as the positive integers.
At 2:53 AM, Rich Hughes said…
That's not what you said originally. Keep changing the definition, chubs, I'll keep finding things wrong. Another 'fail' for you and your theory.
At 7:03 AM, Joe G said…
Yes Richie, I understand that you are too stupid to follow along.
At 7:05 AM, Joe G said…
socle,
What do you mean by "Cantor's method works"?
How does it "work"? What is it used for?
The point is it doesn't work because it does NOTHING.
Are you really that dense?
At 7:06 AM, Joe G said…
Jerad,
Buy a vowel you clueless ass.
At 7:30 AM, Joe G said…
socle,
As I said, with elipsis what happens in the finite extends to the infinite a form of uniformitarianism.
Therefor if one set is greater then the other in the finite world, and that is just extended to the infinite, it will be greater than the other set forever as they both go on and on and on. That set will always have more elements than the other set.
At 8:33 AM, socle said…
socle,
What do you mean by "Cantor's method works"?
How does it "work"? What is it used for?
The point is it doesn't work because it does NOTHING.
Are you really that dense?
As I mentioned before, the notion of "countably infinite" sets is used in measure theory. Measure theory of one of the specialties of Dr^2 Dembski, and this concept is cited it at least some of his ID works.
One related application is this: Every countably infinite subset of the real numbers has Lebesgue measure zero, so if you are doing Lebesgue integration, you could, roughly speaking, change the value of the function on a countably infinite subset of its domain without changing the value of its Lebesgue integral.
More generally, countably infinite sets all have certain properties in common. You can prove theorems that apply to all countably infinite sets, regardless of their JoeC, so it makes sense to treat them as a single category. So if I know that CantorC(S) = alephnull, then I know these theorems will apply to S.
In other words, if I have a set which can be represented as an infinite list, then I automatically know this set will have certain properties, all without having to apply the Einstein Train test or whatever.
At 8:52 AM, Joe G said…
As I mentioned before, the notion of "countably infinite" sets is used in measure theory.
What does that have to do with saying all countably infinite sets have the same cardinality?
At 9:12 AM, socle said…
What does that have to do with saying all countably infinite sets have the same cardinality?
Well at least Dembski says this, implicitly. No one uses the standard term "countably infinite" without assuming this. So this refutes your claim that:
"How does it "work"? What is it used for? The point is it [Cantor's method] doesn't work because it does NOTHING."
I've shown that Cantor's concept of cardinality is at least used to make specious ID arguments.
At 9:23 AM, Joe G said…
No one uses the standard term "countably infinite" without assuming this.
LoL! That's it?
No one uses Cantor's concept of cardinality with respect to countably infinite sets.
And you cannot demonstrate otherwise.
At 11:30 AM, Joe G said…
I've shown that Cantor's concept of cardinality is at least used to make specious ID arguments.
Cantor's is a specious concept. So whoever uses it and you have not shown that Dembski has is, in all likely hood, going to have a specious argument.
At 5:52 PM, Unknown said…
"Buy a vowel you clueless ass."
Well Joe, I'm might be clueless but at least I can carry on a respectful conversation. You're just a bully who isn't good at dealing with dissent.
Let's get back to the basic question. If we take two sets and match them up term for term so that every element in one set is uniquely matched with an element of the other set then are those sets the same size?
No mare name calling, not more bullying. Is a onetoone matching a fair way of comparing sizes of sets or not?
At 5:59 PM, Joe G said…
I can handle dissent. Just repeating what everyone else has already sed is NOT dissent, Jerad. It is being an ass.
And I have already explained everything you have asked. I am not going through it again just because you want to be a wanker.
At 9:21 PM, Joe G said…
Let's get back to the basic question. If we take two sets and match them up term for term so that every element in one set is uniquely matched with an element of the other set then are those sets the same size?
AGAIN:
“It depends on the underlying order relation you use for comparing infinite sets”. Stephan Kulla and JoeG
What good is your MS in math if you can't even follow a discussion?
At 2:21 AM, Unknown said…
“It depends on the underlying order relation you use for comparing infinite sets”.
Alright. Could you define 'order relation'? Could you give a few examples of how a different order relation changes the size comparison.
Also . . . just to check . . . .
If A = {1, 2, 3, 4, . . . . } it's cardinality is infinity.
If B = {2, 3, 4, 5, . . . } it's cardinality is also infinity but a smaller infinity than A's? Or is it iniinity minus one?
And C = {2, 4, 6, 8 . . . } is also infinitely large but smaller than A or B. And it's cardinality is what exactly? A's infinity divided by 2?
Is there a smallest infinity? Which one is it? If there is and you subtract one from it is it no longer infinity?
If D = {1, 3, 5, 7 . . . } then it's infinitely large. Is it's infinity the same size as C's?
Since A = the union of C and D then A's infinity is C's infinity + D's infinity?
How do you define the four basic mathematical operations with your different infinities? What is C's infinity divided by B's infinity?
At 10:01 AM, Joe G said…
Jerad,
Order relation has already been defined. It pits the same numbers against each other.
Ya see Jerad, numbers are not arbitrary objects. they actually mean something and occupy a point along the number line just as I have been saying for days.
Also the ilk from the TSZ linked to a math page that says there are different sizes of infinity.
Yes numbers going to infinity will be really, really, large. But that does NOT make them equal.
Math degreee my ass....
At 12:05 PM, Unknown said…
So, if your infinities are numbers you should be able to do arithmetic with them. And I asked you what some of your rules would be.
I'd still like to hear what you think the cardinality of my set C divided by the cardinality of my set B is.
Or whether or not you think there is a smallest infinity.
The proof that the cardinality of the real numbers is larger than the cardinality of the positive integers is pretty simple. You just have to show that no matter how you try and put the two sets in onetoone correspondence there will always be a real number not on the list. So, there must be more of them. The real numbers are uncountably infinite. Cantor's continuum hypothesis is involved with this issue.
Real numbers do mean 'something' in the real world. But we're talking mathematics here. And some numbers (like the imaginary numbers) end up having applications in the real world which was quite surprising when it was discovered.
Any time you want to have a race to see who can more quickly solve a differential equation using the variation of parameters method let me know.
At 7:43 PM, Joe G said…
No Jerad, dumbass. There is equal to, greater than, less than, greater than or equal to or less than or equal to.
Only a moron would think there would be exact numbers.
Again all Cantor does is through up his hands and "they are such big numbers we will just call them equal"
What maths can we do with that?
At 2:52 AM, Unknown said…
No Cantor did not just throw up his hands and give up. He figured out that there were different sizes of infinity and that meant he had to forget about normal ways of thinking about numbers when moving to the infinite.
He realised that the process of matching up elements from sets on a onetoone basis was valid for compaing sizes for finite as well as infinite sets. This gave the counterintuitive result that the positive integers and the even positive integers were the same 'size'.
When he realised there were different sizes of infinity then he had to deal with ordering those infinities. He decided there was a smallest infinity.
What do you say, is there a smallest infinity? And what do you get if you divide it by two? And divide it by two again? And again? And again? Will you keep getting smaller and smaller infinities? Is there an infinite regress?
In fact, take A = {1, 2, 3, 4, . . . . } and C = {2, 4, 6, 8 . . . } You agree they are both infinite sets and they both have infinite cardinality. But that C is half as big as A.
Take P = {2, 3, 5, 7, 11, 13, 17 . . . } (the set of all the prime numbers). Is the cardinality of P bigger or smaller than the cardinality of C? They have one element in common but that's it.
Compare P to D = {4, 8, 12, 16 . . . } P and D have no elements in common so how can you tell which is bigger?
What about F = {1, 1, 2, 3, 5, 8, 13, 21, . . . . } (the Fibonacci sequence) How does it's cardinality compare with C?
At 9:46 AM, Joe G said…
No Cantor did not just throw up his hands and give up.
Yes, he did.
He figured out that there were different sizes of infinity and that meant he had to forget about normal ways of thinking about numbers when moving to the infinite.
If there are different sizes of infinity then why does infinity = infinity to him?
He realised that the process of matching up elements from sets on a onetoone basis was valid for compaing sizes for finite as well as infinite sets.
Yet he doesn't match elements on an onetoone basis.
In fact, take A = {1, 2, 3, 4, . . . . } and C = {2, 4, 6, 8 . . . } You agree they are both infinite sets and they both have infinite cardinality. But that C is half as big as A.
That's a fact.
Take P = {2, 3, 5, 7, 11, 13, 17 . . . } (the set of all the prime numbers). Is the cardinality of P bigger or smaller than the cardinality of C?
Do the math Mr Math degree.
That you have to ask me says that you are totally clueless....
At 2:21 PM, Unknown said…
"If there are different sizes of infinity then why does infinity = infinity to him?"
Depends on which infinities you're talking about. As discussed in any Set Theory course.
"Yet he doesn't match elements on an onetoone basis."
Well, he certainly did. And so can you: You can match up the postitive integers with the set {2, 4, 6 . . . }
1 <> 2
2 <> 4
3 <> 6
And so on. Each element of each set is uniquely matched with an element of the other set. A onetoone matching/mapping/function.
"Do the math Mr Math degree."
I think I do know the cardinality of the set of all Prime Numbers but I want to know what your method says.
And, I'd like to know, if you think there is a smallest infinity. Something you seem to want to avoid answering for some reason.
At 2:25 PM, Joe G said…
Each element of each set is uniquely matched with an element of the other set. A onetoone matching/mapping/function.
If you have to use a function then it is a given the two sets are of different sizes.
And, I'd like to know, if you think there is a smallest infinity.
I haven't thought about it. But obvioulsy there ARE different sizes of infinity.
As I said to find out what P is wrt C, do the fucking function mapping. You can do that can't you Mr Math bluffer?
At 3:37 PM, Unknown said…
"If you have to use a function then it is a given the two sets are of different sizes."
Really? Even if the function is just 'first element of A matches with first element of B, etc."
" 'And, I'd like to know, if you think there is a smallest infinity. '
I haven't thought about it. But obvioulsy there ARE different sizes of infinity."
So . . . Cantor proposes a smallest infinity, as you would know from your Set Theory course. What say you?
"As I said to find out what P is wrt C, do the fucking function mapping. You can do that can't you Mr Math bluffer?"
Uh no. I can't. I do not know of a function that maps the (let's say positive integers for simplicity) positive integers with the primes. Except to line up the sets. But it doesn't help me predict what the next prime will be.
Just matching the elements of the sets up does not give a numeric scale for the sizes of the infinities.
At 7:27 PM, Joe G said…
Even if the function is just 'first element of A matches with first element of B, etc."
That all depends on what you mean by "match". A 1 matches a 1, not a 2. So if you have to concoct a function to make the 1 map to a 2, then you have already lost.
So . . . Cantor proposes a smallest infinity, as you would know from your Set Theory course. What say you?
If it's possible.
I do not know of a function that maps the (let's say positive integers for simplicity) positive integers with the primes. Except to line up the sets.
So you just line up the sets one above the other and draw lines element by element? That's your methodology? Really?
Just matching the elements of the sets up does not give a numeric scale for the sizes of the infinities.
It should give you relative sizes of the sets. But only if you do it properly.
At 2:54 AM, Unknown said…
"That all depends on what you mean by "match". A 1 matches a 1, not a 2. So if you have to concoct a function to make the 1 map to a 2, then you have already lost."
A mapping can be a formula or just a clear and unabmbiguous series of examples. And the way the mapping is elucidated doesn't affect the sizes of the sets involved.
"So you just line up the sets one above the other and draw lines element by element? That's your methodology? Really?"
That's what I've been saying and doing for many days now. How would you map the positive integers to the primes?
What is the cardinality of the set of primes? My method can answer that. What does your method say?
What is the cardinality of the set of the digits of Pi in order?
What is the cardinality of the set of the digits of e in order?
What is the cardinality of the set of the digits of arctan(Pi)?
What is the cardinality of the set of an unending series of coin flips?
At 9:35 AM, Joe G said…
And the way the mapping is elucidated doesn't affect the sizes of the sets involved.
No, it eluciadtes them.
"So you just line up the sets one above the other and draw lines element by element? That's your methodology? Really?"
That's what I've been saying and doing for many days now.
Yes and it still seems worthless/ useless.
What is the cardinality of the set of primes? My method can answer that.
YOUR method? Or do you mena Cantor's method?
My method says it is infinite and at teh same time still smaller than the set of all nonnegative integers.
At 1:55 PM, Unknown said…
" 'And the way the mapping is elucidated doesn't affect the sizes of the sets involved.'
No, it eluciadtes them."
What? Seriously, that doesnt' make sense.
" 'What is the cardinality of the set of primes? My method can answer that.'
YOUR method? Or do you mena Cantor's method?
My method says it is infinite and at teh same time still smaller than the set of all nonnegative integers."
Okay, Cantor's method (accepted by mathematicians for the last 100 years).
So, how much smaller is the set of all prime numbers? Can you give a quantative answer?
It's a fair question.
At 2:02 PM, Joe G said…
What? Seriously, that doesnt' make sense.
That's because you are incapable of floowing along. As I have been saying the mapping function gives us the relative cardinality.
Okay, Cantor's method (accepted by mathematicians for the last 100 years).
Accepted but not used for anything. LoL!
So, how much smaller is the set of all prime numbers?
Don't know. And YOU even provided the reason. Go figure...
Can you give a quantative answer?
No, and YOU even provided the reason. Go figure.
At 4:45 PM, Unknown said…
"That's because you are incapable of floowing along. As I have been saying the mapping function gives us the relative cardinality."
I am very capable. But the mapping function does not give the relative cardinality.
A = {1, 2, 3, 4 . . . .}
P = {2, 3, 5, 7, 11, 13 . . . } (the prime numbers)
There is no 'formula' for this mapping. But it is well defined, unambiguous and clear.
If your method depends on a 'formula' then it's limited.
"Accepted but not used for anything. LoL!"
From Wikipedia:
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.
Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.
" 'So, how much smaller is the set of all prime numbers?'
Don't know. And YOU even provided the reason. Go figure…"
Ah, but I know the cardinality of the primes is the same as the cardinality of the positive integers. You think 'cause there isn't a formula it can't be determined. Your method is limited to mappings which are defined by a formula. Cantor's are not. Canoto's methods are more powerful and handle more situations.
" 'Can you give a quantative answer?'
No, and YOU even provided the reason. Go figure."
But I can provide a quantitative answer. Go figure.
Your ideas of simple, fourth grade mathematics went out in the 19th century. Time to update. Mathematics is more complicated and interesting than that.
At 6:06 PM, Joe G said…
But the mapping function does not give the relative cardinality.
A = {1, 2, 3, 4 . . . .}
P = {2, 3, 5, 7, 11, 13 . . . } (the prime numbers)
There is no 'formula' for this mapping. But it is well defined, unambiguous and clear.
It's clearly bullshit. All you are doing is saying the numbers are arbitrary elements and their place on the number line is meaningless.
And again you prove that you are incapable of following along. You must be a moron. I am not arguing against set theory, yet you reference set theory.
I am arguing against one little and obviously insignificant part of it. That you cannot understand that proves that you are an asshole.
Ah, but I know the cardinality of the primes is the same as the cardinality of the positive integers.
Except you don't know that.
Your method is limited to mappings which are defined by a formula.
Nope. That just gives the relative cardinality. And saying it is greater than is saying something.
Cantor's methods say that numbers are meaningless. And that doesn't bode very well for math.
At 12:19 AM, Unknown said…
"It's clearly bullshit. All you are doing is saying the numbers are arbitrary elements and their place on the number line is meaningless."
I'm talking about the sizes of two infinite sets. They happen to contain recognizeable number sequences.
"And again you prove that you are incapable of following along. You must be a moron. I am not arguing against set theory, yet you reference set theory."
You are if you think there are more positive integers than positive even integers.
"I am arguing against one little and obviously insignificant part of it. That you cannot understand that proves that you are an asshole."
Profanity is not an argument.
" 'Ah, but I know the cardinality of the primes is the same as the cardinality of the positive integers.'
Except you don't know that."
Yup, I do. Because the set containing the primes can be put into a onetoone correspondence with the set containing the positive integers.
" 'Your method is limited to mappings which are defined by a formula.'
Nope. That just gives the relative cardinality. And saying it is greater than is saying something."
Relative cardinality? There are countably infinite sets (like the integers, the primes, the rational numbers, the multiples of 2, the digits of Pi, etc) and there are uncountably infinite sets (like the real numbers, the transcendental numbers, the irrational numbers, etc). Like quantum levels, sets fit into one class or the other.
"Cantor's methods say that numbers are meaningless. And that doesn't bode very well for math."
Cantor's work is about sets. What's in the sets is not the point.
At 7:22 AM, Joe G said…
They happen to contain recognizeable number sequences.
Number sequences are virtyally meaningless once they are placed into sets. You have proved that to me beyond a shadow of a doubt.
And it is all moot anyway as nested hierarchies don't care about Cantor.
At 7:32 AM, Unknown said…
"Number sequences are virtyally meaningless once they are placed into sets. You have proved that to me beyond a shadow of a doubt."
Good thing you're not a Set Theorist then isn't it?
"And it is all moot anyway as nested hierarchies don't care about Cantor."
I'm not an expert but if they're finite then I suspect that is true. It would certainly depend on the particular application of nested hierarchies though.
How did you get caught up in an argument about infinite sets then? I was only responding to that subtopic.
At 7:36 AM, Joe G said…
How did you get caught up in an argument about infinite sets then?
oleg and the other evoTARDs wanted to show people that I didn't understand nested hierarchies by tyrying to show that I didn't understand one insignificant part of set theory.
At 9:54 AM, Unknown said…
" 'How did you get caught up in an argument about infinite sets then?'
oleg and the other evoTARDs wanted to show people that I didn't understand nested hierarchies by tyrying to show that I didn't understand one insignificant part of set theory."
Well, if you got caught up in something that wasn't pertinent then who won?
At 11:16 AM, Joe G said…
Well if the TARDs are forced into irrelevant and dishonest distractions in order to try to in, it is obvious that I had already won.
At 11:29 AM, Unknown said…
"Well if the TARDs are forced into irrelevant and dishonest distractions in order to try to in, it is obvious that I had already won."
Why did you follow their lead then?
At 11:33 AM, Joe G said…
To see how far they would go in their desperation. And obvioulsy they are very, very desperate.
At 4:44 PM, Unknown said…
"To see how far they would go in their desperation. And obvioulsy they are very, very desperate."
And so your arguing for your method of determing the cardinality of sets was just to tease everyone? Your post called "How to Determine the Cardinality of sets that go to infinity" was just a smoke screen? Same with "Density vs. Cardinality"?
Post a Comment
<< Home